{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T19:23:03Z","timestamp":1777490583279,"version":"3.51.4"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T00:00:00Z","timestamp":1755734400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T00:00:00Z","timestamp":1755734400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12471296"],"award-info":[{"award-number":["12471296"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Sichuan Science and Technology Program","award":["2025ZNSFSC0813"],"award-info":[{"award-number":["2025ZNSFSC0813"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s10589-025-00719-3","type":"journal-article","created":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T07:16:12Z","timestamp":1755760572000},"page":"121-156","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Accelerated forward-backward algorithms with subgradient corrections"],"prefix":"10.1007","volume":"93","author":[{"given":"Xin","family":"He","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1831-8039","authenticated-orcid":false,"given":"Yaping","family":"Fang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,8,21]]},"reference":[{"issue":"1\u20132","key":"719_CR1","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s10107-018-1350-9","volume":"180","author":"V Apidopoulos","year":"2020","unstructured":"Apidopoulos, V., Aujol, J.F., Dossal, C.: Convergence rate of inertial Forward-Backward algorithm beyond Nesterov\u2019s rule. Math. Program. 180(1\u20132), 137\u2013156 (2020)","journal-title":"Math. Program."},{"issue":"5","key":"719_CR2","doi-asserted-by":"publisher","first-page":"1985","DOI":"10.4171\/jems\/1231","volume":"25","author":"H Attouch","year":"2023","unstructured":"Attouch, H., Bo\u0163, R.I., Csetnek, E.R.: Fast optimization via inertial dynamics with closed-loop damping. J. Eur. Math. Soc. 25(5), 1985\u20132056 (2023)","journal-title":"J. Eur. Math. Soc."},{"key":"719_CR3","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s10107-020-01591-1","volume":"193","author":"H Attouch","year":"2022","unstructured":"Attouch, H., Chbani, Z., Fadili, J., Riahi, H.: First-order optimization algorithms via inertial systems with Hessian driven damping. Math. Program. 193, 113\u2013155 (2022)","journal-title":"Math. Program."},{"issue":"5","key":"719_CR4","doi-asserted-by":"publisher","first-page":"1199","DOI":"10.1080\/02331934.2021.2009828","volume":"72","author":"H Attouch","year":"2023","unstructured":"Attouch, H., Chbani, Z., Fadili, J., Riahi, H.: Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping. Optimization 72(5), 1199\u20131238 (2023)","journal-title":"Optimization"},{"key":"719_CR5","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/s10107-016-0992-8","volume":"168","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Chbani, Z., Peypouquet, J., Redont, P.: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity. Math. Program. 168, 123\u2013175 (2018)","journal-title":"Math. Program."},{"issue":"3","key":"719_CR6","doi-asserted-by":"publisher","first-page":"1824","DOI":"10.1137\/15M1046095","volume":"26","author":"H Attouch","year":"2016","unstructured":"Attouch, H., Peypouquet, J.: The rate of convergence of Nesterov\u2019s accelerated forward-backward method is actually faster than [CDATA[1\/k^2]]$$1\/k^2$$. SIAM J. Optim. 26(3), 1824\u20131834 (2016)","journal-title":"SIAM J. Optim."},{"issue":"10","key":"719_CR7","doi-asserted-by":"publisher","first-page":"5734","DOI":"10.1016\/j.jde.2016.08.020","volume":"261","author":"H Attouch","year":"2016","unstructured":"Attouch, H., Peypouquet, J., Redont, P.: Fast convex optimization via inertial dynamics with Hessian driven damping. J. Differential Equations 261(10), 5734\u20135783 (2016)","journal-title":"J. Differential Equations"},{"issue":"2","key":"719_CR8","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s00245-023-09997-x","volume":"88","author":"H Attouch","year":"2023","unstructured":"Attouch, H., Balhag, A., Chbani, Z., Riahi, H.: Accelerated gradient methods combining Tikhonov regularization with geometric damping driven by the Hessian. Appl. Math. Optim. 88(2), 29 (2023)","journal-title":"Appl. Math. Optim."},{"issue":"1","key":"719_CR9","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1137\/17M1114739","volume":"28","author":"H Attouch","year":"2018","unstructured":"Attouch, H., Cabot, A.: Convergence rates of inertial forward-backward algorithms. SIAM J. Optim. 28(1), 849\u2013874 (2018)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"719_CR10","doi-asserted-by":"publisher","first-page":"2408","DOI":"10.1137\/140994964","volume":"25","author":"JF Aujol","year":"2015","unstructured":"Aujol, J.F., Dossal, C.: Stability of over-relaxations for the forward-backward algorithm, application to FISTA. SIAM J. Optim. 25(4), 2408\u20132433 (2015)","journal-title":"SIAM J. Optim."},{"key":"719_CR11","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1007\/s10107-023-01960-6","volume":"204","author":"JF Aujol","year":"2024","unstructured":"Aujol, J.F., Dossal, C., Rondepierre, A.: FISTA is an automatic geometrically optimized algorithm for strongly convex functions. Math. Program. 204, 449\u2013491 (2024)","journal-title":"Math. Program."},{"key":"719_CR12","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2016","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Cham (2016)","edition":"2"},{"issue":"1","key":"719_CR13","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1137\/080716542","volume":"2","author":"A Beck","year":"2009","unstructured":"Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183\u2013202 (2009)","journal-title":"SIAM J. Imaging Sci."},{"issue":"2","key":"719_CR14","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s10589-022-00421-8","volume":"84","author":"Y Bello-Cruz","year":"2023","unstructured":"Bello-Cruz, Y., Gon\u00e7alves, M.L., Krislock, N.: On FISTA with a relative error rule. Comput. Optim. Appl. 84(2), 295\u2013318 (2023)","journal-title":"Comput. Optim. Appl."},{"key":"719_CR15","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10107-020-01528-8","volume":"189","author":"RI Bo\u0163","year":"2021","unstructured":"Bo\u0163, R.I., Csetnek, E.R., L\u00e1szl\u00f3, S.C.: Tikhonov regularization of a second order dynamical system with Hessian driven damping. Math. Program. 189, 151\u2013186 (2021)","journal-title":"Math. Program."},{"key":"719_CR16","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/s10208-023-09636-5","volume":"25","author":"RI Bo\u0163","year":"2025","unstructured":"Bo\u0163, R.I., Csetnek, E.R., Nguyen, D.K.: Fast optimistic gradient descent ascent (OGDA) method in continuous and discrete time. Found. Comut. Math. 25, 163\u2013222 (2025)","journal-title":"Found. Comut. Math."},{"issue":"1","key":"719_CR17","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/s10107-022-01879-4","volume":"200","author":"RI Bo\u0163","year":"2023","unstructured":"Bo\u0163, R.I., Csetnek, E.R., Nguyen, D.K.: Fast augmented Lagrangian method in the convex regime with convergence guarantees for the iterates. Math. Program. 200(1), 147\u2013197 (2023)","journal-title":"Math. Program."},{"key":"719_CR18","doi-asserted-by":"publisher","first-page":"968","DOI":"10.1007\/s10957-015-0746-4","volume":"166","author":"A Chambolle","year":"2015","unstructured":"Chambolle, A., Dossal, C.: On the convergence of the iterates of the \u201cfast iterative shrinkage\/thresholding algorithm.\u201d J. Optim. Theory Appl. 166, 968\u2013982 (2015)","journal-title":"J. Optim. Theory Appl."},{"key":"719_CR19","unstructured":"Chen, L., Luo, H.: First order optimization methods based on Hessian-driven Nesterov accelerated gradient flow. arXiv:1912.09276. (2019)"},{"issue":"4","key":"719_CR20","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1137\/0802032","volume":"2","author":"O G\u00fcler","year":"1992","unstructured":"G\u00fcler, O.: New proximal point algorithms for convex minimization. SIAM J. Optim. 2(4), 649\u2013464 (1992)","journal-title":"SIAM J. Optim."},{"key":"719_CR21","doi-asserted-by":"publisher","first-page":"1669","DOI":"10.1007\/s11075-021-01246-y","volume":"90","author":"X He","year":"2022","unstructured":"He, X., Hu, R., Fang, Y.P.: Inertial accelerated primal-dual methods for linear equality constrained convex optimization problems. Numer. Algorithms 90, 1669\u20131690 (2022)","journal-title":"Numer. Algorithms"},{"key":"719_CR22","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1007\/s00245-023-10102-5","volume":"89","author":"X He","year":"2024","unstructured":"He, X., Hu, R., Fang, Y.P.: A second order primal-dual dynamical system for a convex-concave bilinear saddle point problem. Appl. Math. Optim. 89, 30 (2024)","journal-title":"Appl. Math. Optim."},{"issue":"1","key":"719_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2024.108289","volume":"140","author":"X He","year":"2025","unstructured":"He, X., Huang, N.J., Fang, Y.P.: Non-ergodic convergence rate of an inertial accelerated primal-dual algorithm for saddle point problems. Commun. Nonlinear Sci. Numer. Simul. 140(1), 108289 (2025)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"2","key":"719_CR24","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1080\/02331934.2023.2253813","volume":"74","author":"X He","year":"2025","unstructured":"He, X., Tian, F., Li, A.Q., Fang, Y.P.: Convergence rates of mixed primal-dual dynamical systems with Hessian driven damping. Optimization 74(2), 365\u2013390 (2025)","journal-title":"Optimization"},{"key":"719_CR25","doi-asserted-by":"crossref","unstructured":"Li, B., Shi, B., Yuan, Y.X.: Proximal subgradient norm minimization of ISTA and FISTA. arXiv:2211.01610. (2022)","DOI":"10.2139\/ssrn.4295674"},{"key":"719_CR26","doi-asserted-by":"publisher","first-page":"929","DOI":"10.1007\/s10107-021-01721-3","volume":"195","author":"T Lin","year":"2022","unstructured":"Lin, T., Jordan, M.I.: A control-theoretic perspective on optimal high-order optimization. Math. Program. 195, 929\u2013975 (2022)","journal-title":"Math. Program."},{"key":"719_CR27","unstructured":"Jordan, M.I.: Dynamical, symplectic and stochastic perspectives on gradient-based optimization. In Proceedings of the International Congress of Mathematicians: Rio de Janeiro. 523-549 (2018)"},{"key":"719_CR28","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/s10107-015-0949-3","volume":"159","author":"D Kim","year":"2016","unstructured":"Kim, D., Fessler, J.A.: Optimized first-order methods for smooth convex minimization. Math. Program. 159, 81\u2013107 (2016)","journal-title":"Math. Program."},{"issue":"1\u20132","key":"719_CR29","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1007\/s10107-021-01713-3","volume":"195","author":"H Luo","year":"2022","unstructured":"Luo, H., Chen, L.: From differential equation solvers to accelerated first-order methods for convex optimization. Math. Program. 195(1\u20132), 735\u2013781 (2022)","journal-title":"Math. Program."},{"issue":"4","key":"719_CR30","doi-asserted-by":"publisher","first-page":"1029","DOI":"10.1007\/s10898-019-00808-8","volume":"75","author":"RD Mill\u00e1n","year":"2019","unstructured":"Mill\u00e1n, R.D., Machado, M.P.: Inexact proximal $$\\epsilon $$[CDATA[\\epsilon ]]-subgradient methods for composite convex optimization problems. J. Global Optim. 75(4), 1029\u20131060 (2019)","journal-title":"J. Global Optim."},{"issue":"2","key":"719_CR31","doi-asserted-by":"publisher","first-page":"1092","DOI":"10.1137\/110833786","volume":"23","author":"RD Monteiro","year":"2013","unstructured":"Monteiro, R.D., Svaiter, B.F.: An accelerated hybrid proximal extragradient method for convex optimization and its implications to second-order methods. SIAM J. Optim. 23(2), 1092\u20131125 (2013)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"719_CR32","first-page":"372","volume":"27","author":"Y Nesterov","year":"1983","unstructured":"Nesterov, Y.: A method of solving a convex programming problem with convergence rate $${\\cal{O} }(1\/k^2)$$[CDATA[{\\cal{O} }(1\/k^2)]]. Sov. Math. Dokl. 27(2), 372\u2013376 (1983)","journal-title":"Sov. Math. Dokl."},{"key":"719_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-91578-4","volume-title":"Lectures on Convex Optimization","author":"Y Nesterov","year":"2018","unstructured":"Nesterov, Y.: Lectures on Convex Optimization. Springer, Berlin (2018)"},{"key":"719_CR34","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14, 877\u2013898 (1976)","journal-title":"SIAM J. Control. Optim."},{"key":"719_CR35","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/s10107-021-01681-8","volume":"195","author":"B Shi","year":"2022","unstructured":"Shi, B., Du, S.S., Jordan, M.I., Su, W.J.: Understanding the acceleration phenomenon via high-resolution differential equations. Math. Program. 195, 79\u2013148 (2022)","journal-title":"Math. Program."},{"key":"719_CR36","unstructured":"Shi, B., Du, S.S., Su, W.J., Jordan, M.I.: Acceleration via symplectic discretization of high-resolution differential equations. Adv. Neural. Inf. Process. Syst. 5745-5753 (2019)"},{"issue":"153","key":"719_CR37","first-page":"1","volume":"17","author":"W Su","year":"2016","unstructured":"Su, W., Boyd, S., Cand\u00e8s, E.: A differential equation for modeling Nesterov\u2019s accelerated gradient method: theory and insights. J. Mach. Learn. Res. 17(153), 1\u201343 (2016)","journal-title":"J. Mach. Learn. Res."},{"issue":"3","key":"719_CR38","doi-asserted-by":"publisher","first-page":"1607","DOI":"10.1137\/110844805","volume":"23","author":"S Villa","year":"2013","unstructured":"Villa, S., Salzo, S., Baldassarre, L., Verri, A.: Accelerated and inexact forward-backward algorithms. SIAM J. Optim. 23(3), 1607\u20131633 (2013)","journal-title":"SIAM J. Optim."},{"key":"719_CR39","doi-asserted-by":"publisher","first-page":"E7351","DOI":"10.1073\/pnas.1614734113","volume":"113","author":"A Wibisono","year":"2016","unstructured":"Wibisono, A., Wilson, A.C., Jordan, M.I.: A variational perspective on accelerated methods in optimization. Proc. Natl. Acad. Sci. U.S.A. 113, E7351\u2013E7358 (2016)","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"issue":"113","key":"719_CR40","first-page":"1","volume":"22","author":"AC Wilson","year":"2021","unstructured":"Wilson, A.C., Recht, B., Jordan, M.I.: A Lyapunov analysis of accelerated methods in optimization. J. Mach. Learn. Res. 22(113), 1\u201334 (2021)","journal-title":"J. Mach. Learn. Res."},{"key":"719_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/s40305-024-00542-3","author":"YX Yuan","year":"2024","unstructured":"Yuan, Y.X., Zhang, Y.: Analyze accelerated mirror descent via high-resolution ODEs. J. Oper. Res. Soc. China. (2024). https:\/\/doi.org\/10.1007\/s40305-024-00542-3","journal-title":"J. Oper. Res. Soc. China."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00719-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-025-00719-3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00719-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T14:26:06Z","timestamp":1767968766000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-025-00719-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,21]]},"references-count":41,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["719"],"URL":"https:\/\/doi.org\/10.1007\/s10589-025-00719-3","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,21]]},"assertion":[{"value":"4 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}